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# st: What is the rationale of the covariance estimator using by Brant test?

 From To Subject st: What is the rationale of the covariance estimator using by Brant test? Date Mon, 19 Nov 2012 00:12:50 -0000

```Dear all,
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```
I would like to know the rationale of the asymptotic covariance matrix of the Brant test. Consider that there is a 3-category dependent variable z. To do Brant test, we dichotomise z as two variable, zj, and zl. There two variables are separately modelled by two binary logit models, j and l. Let bj is the MLEs of the model j, and bj the MLEs of the model l. Brant presented an estimator for the covariance between bj and bl (X is the matrix of independent variables with a constant vector. Wjl = diag(πl-πjπl), where πj is the predicted probability of zj=1 estimated by the model j, and πl is Pr(yl=1) estimated by the model l.)
```.
COV(bj,bl) = [(X’ Wjj X)^-1] [X’ Wjl X] [(X’ Wll X)^-1]
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```
Where is this estimator derived from? I know that, if we use this estimator to compute cov(bj, bj), then the equation above shrinks back to
```.
COV(bj,bj) = (X’ Wjj X)^-1
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```
which is just the covariance matrix of binary logit model j. However, when we compute COV(bj,bl), the estimator looks like a “sandwich”:
```.
COV(bj,bl) = COV(bj,bj) [X’ Wjl X] COV(bl,bl)
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```
The first and the third term in the right-hand side are the covariance matrices of the model j and l respectively. But, my questions are: (1) where does the middle term in the right-hand side come from? (2) why should the three terms in the right-hand side be put together in that way?
```.
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I have compared the covariance computed by Brant’s estimator with that computed by the command “suest”. The results were different. So Brant’s estimator cannot be explained by the theory of the seemingly unrelated estimation. (If they are indeed theoretically different, then which estimator is better?) Moreover, I have compared the results computed by Brant’s estimator and by gologit2. They are also different. (Specifically, I restricted the coefficients of gologit2 to b1 and b2, and then compared the e(V) of the gologits with the covariance computed by Brant’s estimator.) So Brant’s estimator cannot be directly explained by the way that the generalized ordered logit estimates the covariance.
```.
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I’d appreciate it if someone can explain the rationale of Brant’s covariance estimator. (PS: I know how to compute it. I want to know how it comes from.) Thank you very much.
```.
The paper of the Brant test is:
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Brant, Rollin. 1990. "Assessing Proportionality in the Proportional Odds Model for Ordinal Logistic Regression." Biometrics 46(4): 1171-1178.
```https://docs.google.com/open?id=0B984NoKuZv46WUdlZlZHbmliV2s
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```
Chi-lin Tsai
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```